Insights into time fractional dynamics in the Belousov-Zhabotinsky system through singular and non-singular kernels

In the realm of nonlinear dynamics, the Belousov-Zhabotinsky reaction system has long held the fascination of researchers. The Belousov-Zhabotinsky system continues to be an active area of research, offering insights into the fundamental principles of nonlinear dynamics in complex systems. To deepen our understanding of this intricate system, we introduce a pioneering approach to tackle the time fractional Belousov-Zhabotinsky system, employing the Caputo and Atangana-Baleanu Caputo fractional derivatives with the double Laplace method. The solution we obtained is in the form of series which helps in investigating the accuracy of the proposed method. The primary advantage of the proposed technique lies in the low amount of calculations required and produce high degree of precision in the solutions. Furthermore, the existence and uniqueness of the solution are investigated thereby enhancing the overall credibility of our study. To visually represent our results, we present a series of 2D and 3D graphical representations that vividly illustrate the behavior of the model and the impact of changing the fractional order derivative and the time on the obtained solutions.


Fractional derivatives
As we mentioned earlier, there are many definitions of fractional differentiation, each of them has its own advantages and disadvantages.It's important to highlight that Caputo's definition is applicable solely to functions that are differentiable.In 2016 Abdon Atangana and Dumitru Baleanu presented a new fractional derivative with non-local and no-singular kernel that depend on Mittag-Leffler function 31 .The studies conducted in recent years following these advancements clearly indicate that scientists have a significant opportunity to address a variety of issues using fractional derivatives.
In this research, our objective is to solve TFBZS using DLM in the sense of Caputo and ABC fractional derivative.Definition 2.1 2 : Caputo fractional derivative is defined as:  p−θ represents the RL fractional integral in the form: Ŵ(.) is the known Gamma function.

Adomian polynomials
The Adomian decomposition technique has introduced the concept that the unknown linear function Q can be represented through a sequence of decompositions: The elements Q j can be recursively calculated, and the nonlinear term F (Q) , which could include expressions like Q 2 , Q 3 , sin Q, exp(Q) , etc. can be represented using Adomian polynomials (AP) denoted as A j within the structure: The calculation of AP is used to handle different forms of nonlinearity.Adomian 37 introduced a technique for computing AP , which has been formally validated.Other methods based on Taylor series have also been developed, as discussed in 38,39 .To compute the AP , A j for the nonlinear term F (Q) , you can apply the follow- ing general formula: Expression (14) can be expanded as follows: (5) From the relations presented in (15), we notice that A 0 depends only on

Framework of Double Laplace method
The double Laplace transform method serves as a valuable mathematical tool for addressing fractional nonlinear equations or systems of equations.This technique proves particularly effective when dealing with equations featuring Caputo, Caputo-Fabrizio, or Atangana-Baleanu-Caputo fractional derivatives.By applying the Laplace transform twice, it enables the conversion of intricate fractional differential equations into more accessible algebraic forms.This transformation simplifies the process of solving fractional differential equations.Moreover, it can be combined with methods like Adomian polynomials, see 32,40 to effectively handle nonlinear terms within these equations.This approach significantly enhances our capability to analyze and solve real-world problems across various scientific and engineering domains.
Definition 3.1 32 The expression for the Double Laplace transform using Caputo fractional derivative when p − 1 < θ ≤ p can be described as follows: Definition 3.2 32 The expression for the Double Laplace transform using Atangana-Baleanu-Caputo fractional derivatives when p − 1 < θ ≤ p can be described as follows: for p = 1, 2, 3, . ...

Analysis of the existence and uniqueness of the solution
In this section, we will establish the existence and the uniqueness of the TFBZS within the context of the ABC sense.To do so, let's rewrite the couple sytem (1) in the following form: Constrained by: apply ABC fractional integral (Definition 2.2) to both sides of the system equations (20), where K 1 and K 2 represent the right hand sides of the system, actually, they called the kernels K 1 (x, t, p) and K 2 (x, t, w) , for simplicity we will write K 1 (p) and K 2 (w).Assume that p(x, t) and w(x, t) have an upper bound if the kernels K 1 (p) and K 2 (w) satisfy the Lipschitz condition, hence The subsequent iterative formulas for p(x, t) and w(x, t) are formulated: ABC D θ t p = F(x, t, p, w), ABC D θ t w = G(x, t, p, w).
p(x, 0) = p 0 (x, t), w(x, 0) = w 0 (x, t).Proof From the existence of equation ( 26) as a solution os the first equation of the proposed system, assume that: then By assuming that p(x, t) is bounded function, apply recursive method for Eq. ( 28), we get ( 23) www.nature.com/scientificreports/Using relation (33) with the inequality (32), we obtain: At t = t 0 , Eq. ( 34) becomes: Similarily, we can show that, where H r (x, t) = w(x, t) − w(x, 0) .To prove the uniqueness of the solution, consider that p(x, t) has two solu- tions p(x, t) and q(x, t) , hence Putting norm on both sides of Eq. ( 37), Thus, ).The same conclusion can be drawn for w(x, t) .

Solution of the TFBZS using DLM
In this section, the DLM is applied to the TFBZS to find approximate solutions, the system will be investigated under two types of initial conditions 22 .
Precise and estimated solutions for the TFBZS at θ = 1 , t = 0.01 , δ = 1 and = 1.5.Apply the DL formula ( 19) into both sides of the system (63), and perform the same steps presented in Case I to obtain the following results: (66) 24δŴ(θ + 1) p 0 = p(x, 0), The final approximate two iterations series solution using ABC fractional derivative for DLM for Case II will be: Table 4 represents the precise, approximate and the absolute error results from solving the TFBZS (case II) using ABC fractional derivative at θ = 1 , t = 0.01 , δ = 3 and = 2 for various x values.
Considering the results we obtained from solving the two cases of initial conditions with different definitions, we notice that the Caputo results are very close to that results for Atangana-Baleanu-Caputo when the (69) w 0 = w(x, 0), fractional order derivative θ = 1 .Therefore, Table 5 illustrates a comparison for the results we obtained for the two cases when θ < 1.

Graphic representations
Graphic representations offer a visual context that enhances the comprehension of data and results.They provide an immediate and intuitive understanding of the relationships, and patterns present in the data, making it easier for researchers and readers to grasp the significance of the findings.Two and three dimensions graphs for the obtained solutions are presented to visualize the behavior of the TFBZS using two different cases of initial conditions each case was dealt with in two definitions of fractional calculus C and ABC .Figure 1 represents the two-dimensional visualization at different fractional order parameter values θ with fixed time t = 0.5 , and at θ = 1 for several steps of time, this in for the first case of initial conditions using DL Caputo.Figure 2 shows the three-dimensional approximate and exact representation for case I using C , the graphs shows great coincides between exact and estimated solution which reflect the efficiency of the used method.Figure 3 clarify the 2D solution of the TFBZS (case I) using ABC DLM for the two unknown functions p and w at different values of θ with fixed time and at several stages of time with fixed θ = 1 .Figure 4 shows the approximate solution in three-dimensions of the TFBZS at δ = 1 and = 1.5 .The graphs using C and ABC also very close to each other, this means that either using C DLM or ABC DLM, we obtain high solution accuracy.Figure 5 shows the estimated solution of using DLM using C for case II of initial conditions.Figure 6 represents the exact and approximate solution in three-dimensions for case II, its clear that, the estimated solution is nearly the exact solution at the same values of parameters.Figures 7 and 8 represent the obtained solution in two-and three-dimensions using ABC DLM for case II of initial conditions.All the represented graphs show coincides between the estimated and exact solution which ensures the validity of the DLM for solution.

Conclusion
In this study, we obtain an approximate series solution for the TFBZS using DLM under varying initial conditions.Each initial condition was examined using both Caputo and Atangana-Baleanu Caputo fractional derivatives.The results obtained showcased an impressive level of accuracy, with errors consistently maintained at a remarkably low magnitude.Furthermore, we conducted a thorough investigation into the existence and uniqueness aspects of the solution, establishing a robust foundation for the validity of our approach.In order to understand the behavior of the solution, we present two and three dimensional graphs to show the impact of the time and the  For the future directions, we envision expanding the application of the DLM to other complex fractional systems with higher-order fractional derivatives that analyze a real world applications.This can involve exploring the applicability of the method in multi-dimensional fractional systems.

Use of AI tools declaration
The authors confirm that they did not utilize any Artificial Intelligence (AI) tools in the development of this article.

Figure 6 .
Figure 6.The exact and approximate solution of the TFBZS (Case II) under C fractional derivative presented in Eq. (67) at δ = 2 and = 2.

Table 5
derivative of the solution.The graphs of approximate and exact solution demonstrate a close resemblance, indicating the accuracy of the obtained solutions.
. A comparison between the results obtained in the two cases at θ = 0.8 , t = 0.01 , δ = 2 and = 3.fractional